Cancer is a worldwide burden in every aspect: physically, emotionally, and financially. A need for innovation in cancer research has led to a vast interdisciplinary effort to search for the next breakthrough. Mathematical modeling allows for a unique look into the underlying cellular dynamics and allows for testing treatment strategies without the need for clinical trials. This dissertation explores several iterations of a dendritic cell (DC) therapy model and correspondingly investigates what each iteration teaches about response to treatment.

In Chapter 2, motivated by the work of de Pillis et al. (2013), a mathematical model employing six ordinary differential (ODEs) and delay differential equations (DDEs) is formulated to understand the effectiveness of DC vaccines, accounting for cell trafficking with a blood and tumor compartment. A preliminary analysis is performed, with numerical simulations used to show the existence of oscillatory behavior. The model is then reduced to a system of four ODEs. Both models are validated using experimental data from melanoma-induced mice. Conditions under which the model admits rich dynamics observed in a clinical setting, such as periodic solutions and bistability, are established. Mathematical analysis proves the existence of a backward bifurcation and establishes thresholds for R0 that ensure tumor elimination or existence. A sensitivity analysis determines which parameters most significantly impact the reproduction number R0. Identifiability analysis reveals parameters of interest for estimation. Results are framed in terms of treatment implications, including effective combination and monotherapy strategies.

In Chapter 3, a study of whether the observed complexity can be represented with a simplified model is conducted. The DC model of Chapter 2 is reduced to a non-dimensional system of two DDEs. Mathematical and numerical analysis explore the impact of immune response time on the stability and eradication of the tumor, including an analytical proof of conditions necessary for the existence of a Hopf bifurcation. In a limiting case, conditions for global stability of the tumor-free equilibrium are outlined.

Lastly, Chapter 4 discusses future directions to explore. There still remain open questions to investigate and much work to be done, particularly involving uncertainty analysis. An outline of these steps is provided for future undertakings.

Mosquitoes are the greatest killers of mankind, and diseases caused by mosquitoes continue to induce major public health and socio-economic burden in many parts of the world (notably in the tropical sub-regions). This dissertation contributes in providing deeper qualitative insights into the transmission dynamics and control of some mosquito-borne diseases of major public health significance, such as malaria and dengue. The widespread use of chemical insecticides, in the form of long-lasting insecticidal nets (LLINs) and indoor residual spraying, has led to a dramatic decline in malaria burden in endemic areas for the period 2000-2015. This prompted a concerted global effort aiming for malaria eradication by 2040. Unfortunately, the gains recorded are threatened (or not sustainable) due to it Anopheles resistance to all the chemicals embedded in the existing insecticides. This dissertation addresses the all-important question of whether or not malaria eradication can indeed be achieved using insecticides-based control. A novel mathematical model, which incorporates the detailed Anopheles lifecycle and local temperature fluctuations, was designed to address this question. Rigorous analysis of the model, together with numerical simulations using relevant data from endemic areas, show that malaria elimination in meso- and holo-endemic areas is feasible using moderate coverage of moderately-effective and high coverage of highly-effective LLINs, respectively. Biological controls, such as the use of sterile insect technology, have also been advocated as vital for the malaria eradication effort. A new model was developed to determine whether the release of sterile male mosquitoes into the population of wild adult female Anopheles mosquito could lead to a significant reduction (or elimination) of the wild adult female mosquito population. It is shown that the frequent release of a large number of sterile male mosquitoes, over a one year period, could lead to the effective control of the targeted mosquito population. Finally, a new model was designed and used to study the transmission dynamics of dengue serotypes in a population where the Dengvaxia vaccine is used. It is shown that using of the vaccine in dengue-naive populations may induce increased risk of severe disease in these populations.